Therefore, this gives a characterization of subcomplexes in a cw complex. A g cw complex y, b is a subcomplex of x, a if y is a gsubspace of x, bis a closed gsubspace of a, and y n y\x n in the cw decomposition. The algebraic properties of certain tensorproduct chain complexes are used to calculate the homology groups of regular coverings of. Allen hatcher, topology of cell complexes pdf in algebraic topology.
The c stands for closurefinite, and the w for weak topology an ndimensional closed cell is the image of an ndimensional closed ball under an attaching map. Suppose that b, f is a pair of spaces of the homotopy type of countable cw complexes. The following is an important class of subcomplexes of a cw complex x. Whitehead torsion, part ii lecture 4 september 9, 2014 in this lecture, we will continue our discussion of the whitehead torsion of a homotopy equivalence f. A nite qcw complex can be obtained by attaching one qcell at each time. For any cwcomplex xthe nskeleton xn consisting of all cells of dimension nis a subcomplex by property 2. Basic homological aspects of cw complexes 3 2 the space xis the union s n 1 x n endowed with the weak topology. A nite cwcomplex is a cwcomplex having only a nite number of cells in that case, x x n for some n2n. In the rst cwstructure on s nabove sn 1 is a subcomplex of s and snis a subcomplex of s1. The requirement that each copy of is an induced subcomplex of i means that is in fact a simplicial complex because faces with the same underlying vertex set will be identi ed. Its cw complex topology is the same as the topology induced from x, as one sees. This addresses the modified question in jeremys comments, on keeping the preferred cwstructure. Given a cw complex x, a subcomplex of xis a subspace a.
In particular, if a is a subcomplex of x, then it is a relative cw complex. For example the suspension of a cwcomplex itself carries the structure of a cwcomplex. A finite cell complex satisfies both conditions c and w. X characteristic maps, for in an arbitrary index set and n 2. Convention in this talk, x y is always taken to have the product topology, so \x y is a cw complex means \the product topology on x y is the same as the weak topology. Sq1 is the \bockstein homomorphism, the connecting homomor phism in the long exact sequence that arises from 0. A compact subspace of a cw complex is contained in a finite subcomplex hot network questions is it ethical to have two undergraduate researchers in the same group compete against one another for leadershipcredit of a research study.
Xis a subcomplex if and only if it is a closed subspace and the disjoint reunion of cells of x. Cwcomplex k, then x is dominated by a countable subcomplex of k. The cwcomplex is naturally expressed as a triangulation of the simplex of nonnegative real vectors 2 rr with pr i1 i. Thus, we construct a di erent functorial factorization in theorem 2. For a simple example, the circle may be regarded as a cwcomplex by attaching two ends of a closed interval to a basepoint. Show that a cw complex is contractible if it is the union of an. I dont know of any particularly natural condition on maps between arbitrary cwcomplexes that will guarantee the image is a cwcomplex other than just directly demanding that the image of. If x is a space, then a subspace of x is a subset a. So one answer to your question could be, compute invariants that are defined using the cwstructure. Then a special point 0 i2yis called free in yif there is a neighborhood of 0 iin yhomeomorphic to ekg ifor some k2z 0 and nite group g i. The simplest cw complex, complex, regular complex, and simplicial complex homeomorphic to the torus. More generally, a cell complex each point of which is contained in some finite subcomplex is a cwcomplex. It is a compacti cation of the con guration space bx.
A subcomplex a of a cw complex x is a subspace which is a union of cells of x, such that if en. X y between topological spaces x and y is a function which is continuous. Morse theory gives an easy proof that a smooth compact manifold is homotopy equivalent to a cwcomplex with nitely many cells. The integral cohomology of the hilbert scheme of two points burt totaro for a complex manifold xand a natural number a, the hilbert scheme xa also called the douady space is the space of 0dimensional subschemes of degree ain x.
Then the complex obtained by identifying the two copies of or \gluing together 1 and 2 along is cohenmacaulay. X which is a union of cells of x, such that the closure of each cell in a is. Problem 10 of masseys list of 1955 6 poses the ques tion of when b, f is homotopically equivalent i. Also, x y has a cw complex structure with open cells of the form e fwhere e is an open cell of xand fis an open cell of y. Y is continuous if and only if fjei is continuous for all b. It is important to notice that for any cw complex xand any n2n, the nskeleton xnis a subcomplex of x. Remark origin of the cw terminology the terminology cwcomplex goes back to john henry constantine whitehead and see the discussion in hatcher, topology of cell complexes, p. A subcomplex of x is a cw complex on its own right. The spin of a pair of cw complexes, one a subcomplex of the other, is defined. Then the subspace ais contained in a nite cw subcomplex of x.
A separable manifold failing to have the homotopy type of a. Y is called cellular if it sends xk to yk for all k. Whitehead torsion, part ii lecture 4 harvard university. Xthat is a cw complex itself and its cells are some of the cells of x. Embeddingof a finite cwcomplex in a sphere springerlink. Thus for each cell in a, the image of its attaching map is contained in a, so a is itself a cw complex. Then xais a cwcomplex with cells corresponding to the cells of xthat are not in a, plus an additional 0cell corresponding to aitself collapsed to a point. A nite complex is a cwcomplex with only nitely many cells. Y with y pathconnected, show that fcan be extended to a map x. Sqi commutes with the connecting homomorphism in the long exact sequence on cohomology. Relative cw complexes if x is obtained from a by iteratively adding cells, then we say a x is a relative cw complex. Standard examples of cwcomplexes include spheres, real or complex projective spaces and. That is, mc is the subcomplex of mconsisting of all simplices with f c, as well as all of their faces.
Its cw complex topology is the same as the topology induced from x, as one sees by noting inductively that. For any cw complex xthe nskeleton xn consisting of all cells of dimension nis a subcomplex by property 2. Cw complexes and basic constructions in this section we. A symmetric cwcomplex is a hausdor topological space xtogether with a partition into. We show that lius characterization for the product k x l to be a cw complex is independent of the usual axioms of set theory.
The words actually is a cwcomplex suggest to me that the cwstructure is known, whereas simply having the homotopy type of a cwcomplex suggests the cwstructure is unknown, or at least not uniquely determined. X the inclusion of a subcomplex into a cw complex, then the pair x,a. S1 to a point is not nullhomotopic by showing that it induces an isomorphism on h 2. In particular the fundamental group of x has to be countable. Then the circle is the disjoint union of its 0cell base point and the complement of the basepoint, which is homeomorphic to the interior of the interval via the characteristic map. A gcw complex y, b is a subcomplex of x, a if y is a gsubspace of x, bis a closed gsubspace of a, and y n y\x n in the cw decomposition. The notation cw comes from the initial letters of the english names for the above two conditions c for closure finiteness and w for weak topology. The integral cohomology of the hilbert scheme of two points. Recall that a space k is a cw complex, if it is a complex with cells ea such.
Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. A separable manifold failing to have the homotopy type of. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ndimensional counterparts see illustration. Cw complexes soren hansen this note is meant to give a short introduction to cw complexes. X, we can define xe to be the subcomplex that corresponds to the intersection of all subcomplexes containing e. In the rst cw structure on s nabove sn 1 is a subcomplex of s and snis a subcomplex of s1. Notation and conventions in the following a space is a topological space and a map f. The real line admits the structure of 1dimensional cw complex with the integers as zerocells and the intervals n. The acyclicity of our resolution reduces to that of a cellular free resolution, supported on, of a related monomial ideal.
X k beahomotopyequivalenceandd beacountabledensesubset of x. We show that lius characterization for the product k x l to be a cwcomplex is independent of the usual axioms of set theory. A cwcomplex is a nice topological space which is, or can be, built up. If x and y are two cw complexes, then a cellular map f.
Moreover, if the manifold does have a boundary, then its g cw complex may be chosen. I dont know of any particularly natural condition on maps between arbitrary cw complexes that will guarantee the image is a cw complex other than just directly demanding that the image of each cell is a union of cells. X which is a union of cells of x, such that the closure of each cell in a is contained in a. This is given by sending a g gcw complex, y y, to the presheaf sending g h gh to y h yh, the subspace of y y fixed by h h see at elmendorfs theorem references. A subcomplex aof a cw complex xis a union of cells ein xsuch that e. A nite q cw complex can be obtained by attaching one qcell at each time. Roughly speaking, a cw complex is made of basic building blocks called cells. Before turning to cw subcomplexes and an adapted class of morphisms, let us. In particular, part 2 of the proof above shows that every subcomplex of a cw complex is a closed subspace of x.
This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation often with a much smaller complex. Namely, c x and c y are each free abelian groups on the open cells of the respective cw complexes. Celllike maps lecture 5 september 15, 2014 in the last two lectures, we discussed the notion of a simple homotopy equivalences between nite cw complexes. Any union and intersection of subcomplexes is a subcomplex. Clearly such an a forms a cw complex with those characteristics maps. What does the quotient space of a cw complex by a subcomplex. Observe that if x is a nite complex, a4 is redundant, since w is the union of the compact sets.
Y is a simple homotopy equivalence depends on the speci ed cell decompositions of xand y. A subcomplex a of a cw complex x is a subspace which is a union of cells of x. For fixed cell structures, unions and intersections of subcomplexes. Exercises 1 cwcomplexes, homotopy extension property. Show that a cw complex retracts onto any contractible subcomplex. X be a closed subspace such that the intersection y. Recall that a complex m is quasifinite81, 82 if every finite subcomplex k of m is contained in a finite subcomplex p such that any map g. By the very definition, a cw complex is given by a space which admits a. Recall that a space k is a cwcomplex, if it is a complex with cells ea such. Suppose that b, f is a pair of spaces of the homotopy type of countable cwcomplexes. The collection of g gcwcomplexes has a full embedding into the infinity,1presheaves on the orbit category orb g orbg. The term cwcomplex comes from \closure nite with the weak topology, where \closure nite refers to a3 and \weak topology refers to a4.
If xand y are cw complexes, it is not hard to see how such tensor products might arise. A simplicial complex is a set of simplices that satisfies the following conditions. The precise definition prescribes how the cells may be topologically glued together. Simplicial cw structures appendix 533 cw complexes with simplicial structures a.
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